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Home » Use the formula for the volume of a cylinder: \( V = \pi r^2 h \). - High Altitude Science

Use the formula for the volume of a cylinder: \( V = \pi r^2 h \). - High Altitude Science

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 where $ t $ is time in years. Find the time $ t $ at which the rate of change of the population is maximized.

📌 We are given the logistic growth function:

📌 To find the time at which the rate of change is maximized, we compute $ P'(t) $, and then find its maximum.

📌 Let’s denote the denominator as $ D(t) = 1 + 9e^{-0.5t} $, so:

📌 P(t) = \frac{1000}{D(t)} \Rightarrow P'(t) = -1000 \cdot \frac{D'(t)}{[D(t)]^2}.

📌 Now compute $ D'(t) $:

📌 D(t) = 1 + 9e^{-0.5t} \Rightarrow D'(t) = 9 \cdot (-0.5)e^{-0.5t} = -4.5e^{-0.5t}.

📌 P'(t) = -1000 \cdot \frac{-4.5e^{-0.5t}}{(1 + 9e^{-0.5t})^2} = \frac{4500e^{-0.5t}}{(1 + 9e^{-0.5t})^2}.

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÷ 12 ≈ 8.33 → next integer is 9 → \(12 \times 9 = 108\)
Largest: 999 ÷ 12 ≈ 83.25 → largest integer is 83 → \(12 \times 83 = 9...
These form an arithmetic sequence: \(108, 120, \dots, 996\) with \(a =...
l = a + (n-1)d \implies 996 = 108 + (n-1) \cdot 12
= (n-1) \cdot 12 \implies n-1 = 74 \implies n = 75
There are \(\boxed{75}\) positive 3-digit numbers divisible by 12.**Qu...
The circular garden is inscribed in a square with side length 10 meter...
\[ r = \frac{d}{2} = \frac{10}{2} = 5 \text{ meters} \]
The area \( A \) of the circle is given by the formula:
\[ A = \pi r^2 = \pi (5)^2 = 25\pi \text{ square meters} \]

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